Transport Energy

Abstract

We introduce the \emph{transport energy} functional $\mathcal{E}$ (a variant of the Bouchitté-Buttazzo-Seppecher shape optimization functional) and we prove that its unique minimizer is the optimal transport density ${\mu }^{\ast }$, i.e., the solution of Monge-Kantorovich equations. We study the gradient flow of $\mathcal{E}$ showing that ${\mu }^{\ast }$ is the unique global attractor of the flow. We introduce a two parameter family $\{{\mathcal{E}}_{\lambda ,\delta }{\}}_{\lambda ,\delta >0}$ of strictly convex functionals approximating $\mathcal{E}$ and we prove the convergence of the minimizers ${\mu }_{\lambda ,\delta }^{\ast }$ of ${\mathcal{E}}_{\lambda ,\delta }$ to ${\mu }^{\ast }$ as we let $\delta \to 0^{+}$ and $\lambda \to 0^{+}.$ We derive an evolution system of fully non-linear PDEs as gradient flow of ${\mathcal{E}}_{\lambda ,\delta }$ in $L^2$, showing existence and uniqueness of solutions. All the trajectories of the flow converge in $W_0^{1,p}$ to the unique minimizer ${\mu }_{\lambda ,\delta }^{\ast }$ of ${\mathcal{E}}_{\lambda ,\delta }.$ Finally, we characterize ${\mu }_{\lambda ,\delta }^{\ast }$ by a non-linear system of PDEs which is a perturbation of Monge-Kantorovich equations by means of a p-Laplacian.